I was wondering whether it would be possible to count number of binary relations possible from a set of $n$ elements (I am mainly interested in the cases where $n=3,4,5,6,7$ but a general formula if possible would be great as well) that are both asymmetric and transitive in nature.
I do know that the possible number of asymmetric relations would be $3^\frac{n^2-n}{2}$ and that there is no method to count number of transitive relations but it doesn't help me much. I was also wondering if counting number of possible strict partial orders would work but I'm not quite sure how to do so, any help in this question is greatly appreciated. Thank you in advance!
Note that asymmetric transitive relations are also irreflexive, thus strict partial orders. Since every strict partial order is a partial order with the "reflexive bits removed", these are in a bijective correspondence, so there's as many of your relations as there are partial orders. Exact, somewhat effcient formulas are unknown as far as I know, but OEIS has a list until $n = 18$.